Particle vs Wave Interpretations of QM

[jbergman]

Do you have a reference for Zurek? I know that he talks about envariance, but I would like to read where he talks about large numbers of worlds.

I haven't had a chance to reread this in detail but I believe this paper by Zurek discusses it, https://doi.org/10.1103/PhysRevA.71.052105

 

[jbergman]

This raises two related questions. First, what does it mean for two branches to be distinct worlds if they differ only infinitesimally?

Two branches are considered distinct if they are decohered. Do you understand that concept? Not sure I can explain it in a few words but essentially branches are distinct when they become entangled with a system with a large number of particles that it is essentially irreversible, i.e., the probability is extremely small that separate branches can recombine. If you are familiar with entangled states someone could explain this on more detail.

Second, how is such a continuum of branches represented within the single universal wavefunction $\Psi_U$ without introducing additional mathematical structure beyond $\Psi_U$ itself?

If the wave function itself is continuous this shouldn't be hard to understand but maybe the discrete case is easier to grok first. It sounds like you don't have much familiarity with some basic concepts here.

 

[WWGD]

No. Its meaning becomes very clear in specific quantum interpretations such as Bohmian mechanics and many worlds.

And the Bohmian Rhpsody.

 

[PeterDonis]

One might imagine that nature somehow generates a multiplicity of branches so that the counting reproduces the Born weights.

That seems to be basically what Carroll is doing in the paper under discussion. And, as I've already pointed out, since it's equivalent to assuming the Born Rule, it's useless as an argument for deriving the Born Rule in the MWI.

 

[PeterDonis]

Two branches are considered distinct if they are decohered.

But on this view, what Carroll is doing in the paper we've been discussing is obviously wrong, since he's claiming that, once decoherence has happened, it might result in multiple "branches" corresponding to a single measurement result, not just one. But there's only one decoherence, and only one term in the entangled wave function corresponding to a given result of the measurement, and on the viewpoint you're taking here (which is also the viewpoint that makes sense to me), that means one branch--not some number of branches that depends on the amplitude of that term in the entangled wave function.

 

[Demystifier]

It always confuses me about how to think about MWI in the Heisenberg picture where the state is just an initial condition and operator's evolve. Is there any literature on the two?

I have a similar question about Bohmian Mechanics. It strikes me as a questionable interpretation if it depends on which picture you use to carry out calculations.

In MWI, the ontic stuff is the state in the Schrodinger picture. In this sense, the Schrodinger picture is a preferred picture.

In Bohmian mechanics it is not so. It is possible to write down the equations in the Heisenberg picture. See e.g. the formalism in my https://arxiv.org/abs/2308.10500 , it turns out that in this formulation Bohmian mechanics becomes even more elegant mathematically, and even though the Heisenberg picture is not used explicitly, the equations are written in such a form that it is easy to see that it does not depend on the picture.

But note also that even the standard QM prefers Schrodinger picture in a certain sense. Namely, the collapse postulate is formulated in the Schrodinger picture. How would you formulate it in the Heisenberg picture? It can be done, but it's rather clumsy and almost nobody uses it.

 

[Roberto Pavani]

Going back to the original wave/particle discussion and to the picture I posted at #102, one can already view the evolution between the slits and the screen as a continuous probability field. If the computation is correct and faithfully represents what is observed experimentally, the density $|\psi|^2$ and the associated probability current form an extended structure in space, not a collection of discrete objects.

If one further assumes that the wavefunction corresponds to something physically real, then this picture appears conceptually closer to de Broglie's pilot wave than to a collection of discrete worlds. I am not claiming that this favors Bohmian mechanics, only that treating the wavefunction as a continuous physical entity naturally evokes de Broglie's pilot-wave intuition.

 

[Demystifier]

And the Bohmian Rhpsody.

Are you referring to the beginning of my https://arxiv.org/abs/physics/0702069 ?

 

[QuarkyMeson]

Going back to the original wave/particle discussion and to the picture I posted at #102, one can already view the evolution between the slits and the screen as a continuous probability field. If the computation is correct and faithfully represents what is observed experimentally, the density $|\psi|^2$ and the associated probability current form an extended structure in space, not a collection of discrete objects.

If one further assumes that the wavefunction corresponds to something physically real, then this picture appears conceptually closer to de Broglie's pilot wave than to a collection of discrete worlds. I am not claiming that this favors Bohmian mechanics, only that treating the wavefunction as a continuous physical entity naturally evokes de Broglie's pilot-wave intuition.

There's physics and then there's red wine physics.

Bohmian mechanics, Everettian mechanics, collapse models, etc all reproduce essentially the same physics. That puts the I in interpretation. Both MWI and BM treat the wavefunction roughly the same way. BM has some baggage about particles. Also the "collection of discrete worlds" isn't a collection of "discrete worlds" probably in the sense you're still thinking about them. Anyway, this is red wine physics, after all. Think about it however you like that makes the math and intuition easier.

 

[Matterwave]

Both MWI and BM treat the wavefunction roughly the same way.

This seems to be a difficult assertion to rectify at face value. BM wave functions simply guide the motions of actual particles while MWI wave functions describe all there is.

But I have heard David Deutsch (strong MWI proponent) say BM is MWI with a bunch of extra stuff. I think a BM proponent (e.g. Tim Maudlin) would very much dispute that haha.

 

[Roberto Pavani]

Please note tha I don't think there are many wolds, but what I think doesnt'matter here.
Since I'm open to any mainstream point of view, I've just asked from the perspective of MWI how many branches there are and if those are real. The answer from all contributors here seems to be yes 2^50 in the example.

 

[Lord Jestocost]

There's physics and then there's red wine physics.

Bohmian mechanics, Everettian mechanics, collapse models, etc all reproduce essentially the same physics. That puts the I in interpretation. Both MWI and BM treat the wavefunction roughly the same way. BM has some baggage about particles. Also the "collection of discrete worlds" isn't a collection of "discrete worlds" probably in the sense you're still thinking about them. Anyway, this is red wine physics, after all. Think about it however you like that makes the math and intuition easier.

Probably, the “I”s feel uneasy with the fundamentally probabilistic world view of quantum mechanics, and an intrinsically deterministic universe is philosophically more appealing to them.

 

[Lord Jestocost]

Please note tha I don't think there are many wolds, but what I think doesnt'matter here.
Since I'm open to any mainstream point of view, I've just asked from the perspective of MWI how many branches there are and if those are real. The answer from all contributors here seems to be yes 2^50 in the example.

From MathPages, entry “Dialogue on Many Worlds” (https://www.mathpages.com/home/kmath701/kmath701.htm)

“Salviati: I disagree Sagredo. MWI merely asserts that all closed systems must evolve according to the Schrödinger equation. MWI is the sole interpretation which allows that to be a complete description.

Sagredo: But unitary evolution according to the Schrödinger equation does not, by itself, constitute an interpretation of quantum mechanics. At the very least, some additional structure would be needed in order to establish a viable mapping from the wave function to the measures of our experience in accord with quantum mechanics – and it is far from clear that any such approach could ever work. If someone orders a statue of David, you cannot simply deliver a block of marble and say “It’s in there”. As John Bell said “the many universes interpretation is a kind of heuristic, simplified theory, which people have done on the backs of envelopes but haven’t really thought through. When you do try to think it though it is not coherent.” He was not talking about philosophical priorities, he was saying MWI does not represent a coherent interpretation on a technical level.”

 

[jbergman]

But on this view, what Carroll is doing in the paper we've been discussing is obviously wrong, since he's claiming that, once decoherence has happened, it might result in multiple "branches", not just one. But there's only one decoherence, and only one term in the entangled wave function corresponding to a given result of the measurement, and on the viewpoint you're taking here (which is also the viewpoint that makes sense to me), that means one branch--not some number of branches that depends on the amplitude of that term in the entangled wave function.

This Zurek paper explains in more detail how you get multiple different decoherent branches, https://arxiv.org/abs/quant-ph/0405161 .

The argument is too complicated for me to reproduce it here without spending more time. But I think the essence is that whenever you do an experiment say measure a QM coin toss with detector you don't just get the decoherence due to the reading on the detector.

There are a large number of other decoherence events happening orthogonal to the experiment so you end up with many branches/worlds. That's where the whole concept on envariance comes from. He then uses this via Schmidt states to "prove" you would get proportion of states in accordance of the Born rule.

This weekend I will try and digest the paper more carefully and write the equations here as that is the clearest way to discuss this.

 

[PeterDonis]

This Zurek paper

Interestingly, this paper claims that its conclusions are independent of any interpretation--in other words, Zurek is claiming to derive the Born Rule period, not just to derive it for the MWI. He uses the "relative state" framework only for convenience.

 

[Roberto Pavani]

TL;DR: A change of basis may simplify the representation of a problem, but the transformed variables generally do not retain the original physical meaning. Therefore, if a swap or similar transformation is introduced, I would expect the associated transformation to remain part of the bookkeeping. Otherwise it is not clear whether the conclusion refers to the original physical system or to a transformed representation of it.

---

I may be misunderstanding the construction, but I have a concern regarding the status of the swap operation.

If a swap is represented by a unitary operator $B$, then the transformed state should be written as

$\psi' = B\psi$.

Starting from the Schrodinger equation

$i\hbar \,\partial_t\psi = H\psi$,

one obtains

$i\hbar \,\partial_t\psi' = BH\psi = BHB^{-1}\psi'$.

Therefore the transformation $B$ becomes part of the bookkeeping and, in a dynamical setting, one would normally also transform the associated operators:

$H' = BHB^{-1}$.

My concern is that the swap seems to be treated as something that can subsequently be ignored once the counting argument is introduced, whereas mathematically the transformation itself should still be present.

A second concern is related to the subsequent grouping of fine-grained components into the same macroscopic outcome. The swap operation is invertible ($B^{-1}=B$, $B^2=I$ for a simple permutation), but the grouping step is not obviously invertible.

For example, if several distinct fine-grained states are later regarded as corresponding to the same outcome and are counted together, information about the finer structure has effectively been discarded. The resulting multiplicity then appears to depend on the chosen fine-graining and subsequent grouping procedure.

For that reason I am not sure that the final counting is an invariant property of the original state itself, rather than a property of a particular representation and grouping of that state.

In particular, the fine-grained states associated with a given outcome appear to function as multiple replicas of the same macroscopic result. If so, the multiplicity seems to be introduced by the construction itself, rather than independently derived from it.

Finally, going from rational multiplicities to irrational Born weights is no longer a counting argument. It becomes a limiting procedure.
Since limiting procedures involving infinite quantities require strict mathematical control, the continuity step is doing nontrivial work and cannot simply be regarded as a straightforward extension of counting.

 

[jbergman]

TL;DR: A change of basis may simplify the representation of a problem, but the transformed variables generally do not retain the original physical meaning. Therefore, if a swap or similar transformation is introduced, I would expect the associated transformation to remain part of the bookkeeping. Otherwise it is not clear whether the conclusion refers to the original physical system or to a transformed representation of it.

---

I may be misunderstanding the construction, but I have a concern regarding the status of the swap operation.

If a swap is represented by a unitary operator $B$, then the transformed state should be written as

$\psi' = B\psi$.

Starting from the Schrodinger equation

$i\hbar \,\partial_t\psi = H\psi$,

one obtains

$i\hbar \,\partial_t\psi' = BH\psi = BHB^{-1}\psi'$.

Therefore the transformation $B$ becomes part of the bookkeeping and, in a dynamical setting, one would normally also transform the associated operators:

$H' = BHB^{-1}$.

My concern is that the swap seems to be treated as something that can subsequently be ignored once the counting argument is introduced, whereas mathematically the transformation itself should still be present.

A second concern is related to the subsequent grouping of fine-grained components into the same macroscopic outcome. The swap operation is invertible ($B^{-1}=B$, $B^2=I$ for a simple permutation), but the grouping step is not obviously invertible.

For example, if several distinct fine-grained states are later regarded as corresponding to the same outcome and are counted together, information about the finer structure has effectively been discarded. The resulting multiplicity then appears to depend on the chosen fine-graining and subsequent grouping procedure.

For that reason I am not sure that the final counting is an invariant property of the original state itself, rather than a property of a particular representation and grouping of that state.

In particular, the fine-grained states associated with a given outcome appear to function as multiple replicas of the same macroscopic result. If so, the multiplicity seems to be introduced by the construction itself, rather than independently derived from it.

There are critiques of this argument similar to this, see https://arxiv.org/abs/quant-ph/0312058?hl=en-US . Not sure how accepted Zurek's arguments are.

 

[Matterwave]

It seems to me, that "deriving the Born rule" is a favorite pastime of MWI folks going all the way back to Everett. And a common criticism seems to be that the derivations are not convincing or are ultimately circular.

For me, the main problem tends to be a conceptual one. It is hard for me to reason why the Born rule *should* or *could* appear in a fundamental, deterministic, theory such as MWI.

Giving the latest Carroll paper as an example, he lost me before I got to his mathematical machinery. Self locating probability, is a subjective/Bayesian view of probability. I have a hard time reconciling this with what I would hope would be an objective fact of the world in MWI -- the branching of the universes. If you go down the frequentist/objective path you hit a bunch of issues with branch counting as has been pointed out repeatedly in this thread.

All that being said, it is not my opinion that the endeavor is hopeless. After all, I would admit subjective probabilities could arise from or be given correspondences to objective ones.

 

[Fra]

Interestingly, this paper claims that its conclusions are independent of any interpretation--in other words, Zurek is claiming to derive the Born Rule period, not just to derive it for the MWI. He uses the "relative state" framework only for convenience.

I personally think the presumed selection of the born rule should follow from some sort of argument that can't possibly be specific to the mwi weirdness. I agree that alot seems highly circular.

I think its part of hte "quantum logic" in the sense of a generalised probability theory. Maybe apreferred theory for rational reasoning involving degree of belief, somewhat analogos to how koglomorov probability can be argued to follow from some "reasonable" requirements of a measure of rational degree of beliefe. For example the cox axioms and follow ups.

Ie. In short it follows from something like; lets assume degree is a real number + add some "reasonable" consistencyt requirments => koglomorov probability.

But the question at hand one is. Assume that instead our degree of belief requires two real numbers, which corresponds perhaps to a merasurement result from two non-commutative "question sets" that we want to "combined" - how? + add some "reasonable" consistency requirekments => can we get quantum logic with born rule?

I'm not aware of a paper that presents it exactly like I would put it, so i'll refer some close, related, nice papers even if Im sure one can pick on many things, from the quantum reconstruction perspective.

"Complex numbers are an intrinsic part of the mathematical formalism of quantum theory, and are perhaps its most mysterious feature. In this paper, we show that the complex nature of the quantum formalism can be derived directly from the assumption that a pair of real numbers is associated with each sequence of measurement outcomes, with the probability of this sequence being a real-valued function of this number pair. By making use of elementary symmetry conditions, and without assuming that these real number pairs have any other algebraic structure, we show that these pairs must be manipulated according to the rules of complex arithmetic. We demonstrate that these complex numbers combine according to Feynman's sum and product rules, with the modulus-squared yielding the probability of a sequence of outcomes."
-- Origin of Complex Quantum Amplitudes and Feynman's Rules

This attracts me in the sense that a pair of real numbers, is more general one one real number. What this means is of course subject to further "interpretation". Personally I see it so that the encoding context (the observer/agent/subsystem can implement either just a simple statistical bayesian picture (entropic flows), with a one-dimensional partition of sample space OR as a two dimensional partition of comlpementary codes. Here a flow between the two partitions might happen, so it likely even allows for non-trivial dynamics beyond entropic style decay etc. Loosely speaking this is how i have always understood it for a long time, expcet of course, the SHARP reconstruction is missing. Many people tried it, but there is always some thing to pick on.

Ariel caticha and et jaynes has written several papers on this as well, decades ago. I recall reading them way back but then my pain issue was on the real numbers introduced. Because some of this are the real weak points in many of these reconstructions IMO. But assumiing that can be fixed, it has some good explanatory ambitions.

/Fredrik

 

[PeterDonis]

I personally think the presumed selection of the born rule should follow from some sort of argument that can't possibly be specific to the mwi weirdness.

If you accept Gleason's Theorem as such an argument (which would be an argument along the same sort of lines that you describe, basically that you want something that can be computed from the wave function and has the properties we want a "probability" to have), that would do it. But any argument along those lines seems to me to require that you believe that measurements have single outcomes. The issue with interpretations like the MWI is that that's not the case--any measurement has all possible outcomes.

 

[Roberto Pavani]

Please note that, in the spirit of the original topic, if electrons are replaced by photons in the double-slit experiment, the description of the region between the slits and the screen is usually expressed in terms of an electromagnetic field distributed over space. This makes it natural to visualize an extended continuous structure in that region. Whether the quantum wavefunction should be regarded in an analogous ontological way is, of course, an interpretive question, but the mathematical description is already field-like before any discussion of branches or worlds is introduced.

 

[PeterDonis]

the description of the region between the slits and the screen is usually expressed in terms of an electromagnetic field distributed over space

Classically, yes, light is treated differently from electrons. But we're not talking about classical physics here.

Quantum mechanically, they're treated the same: they're both fields--quantum fields. (Yes, technically you can use non-relativistic QM for the electrons, not QFT, whereas using NRQM for light is more problematic. But even in NRQM, the wave function is an "extended continuous structure"; electrons aren't point particles.)

 

[Roberto Pavani]

Back to the original wave-versus-particle discussion:
Consider the following modified double-slit experiment.
A narrow vertical aperture is placed at the position of the central constructive-interference maximum and transmits approximately 99% of the probability contained in that fringe. Behind it, a long tube with absorbing side walls allows only particles that remain close to the central axis to reach a final detector.
The setup does not appear to acquire explicit which-path information; it merely selects the central interference region.
What would be expected on the final detector for:

• photons?
• electrons?

Would one observe:

• a narrow central spot?
• diffraction broadening?
• a residual signature of the two-slit origin?
• near nothing (most particles absorbed by the tube walls)?
• something else?

Here the picture:

 

[renormalize]

What would be expected on the final detector for:

• photons?

Why might we expect anything other than the usual single-slit diffraction pattern?

After all, the light passing through your single-slit interference screen 3 is essentially indistinguishable from that which exists due to standard plane-wave illumination of that slit.

 

[Roberto Pavani]

I can have an opinion based on the standard probabilistic interpretation, but I am more interested in the actual experimental outcome than in my expectation.

The standard wave description suggests diffraction after the aperture, for both photons and electrons. Still, seeing what actually happens in such an experiment would be interesting, especially with electrons, because it provides a direct physical test rather than just a theoretical argument.

 

[PeterDonis]

The standard wave description suggests diffraction after the aperture

But you're preventing that by confining everything to a narrow tube after the aperture. In wave terms, the tube would act like a waveguide.

 

[Fra]

If you accept Gleason's Theorem as such an argument (which would be an argument along the same sort of lines that you describe, basically that you want something that can be computed from the wave function and has the properties we want a "probability" to have), that would do it.

Yes that's in the direction! Except the premises in that theorem such assuming that our beliefs form a hilber space hilbert space structure along with the "geometrisation" by assuming that independent means orthogonal are highly non-trivial assumptions. This is a little more than 2 real numbers. Objecting to this is I think why the quantum reconstruction program is still around. I am definitely not happy with Gleasons theorem as the final answer, not because there is something wrong it, but becauyse the premises is part of what I want reconstructed.

I'd say Gleaons theorem is important an but instead of just proving the born measure, indirectly proves how powerful the premises are in the theorem, and they already single out the measure. This result alone is of course great.

So Gleasons theorems might tells us, instead of asking why born rule, we can ask ourselvs, why "belief constructs" should form linear space with certain inner products.

But any argument along those lines seems to me to require that you believe that measurements have single outcomes. The issue with interpretations like the MWI is that that's not the case--any measurement has all possible outcomes.

MWI is weird so Im not quite sure what the purpose is but i see that confusion as a separate question. I dont see that Gleasons theorem needs any actual outrcomes. Given the hilber space and other premises, it defines the measure with those propertis. Wether some "actual measuremnts" are made at all, I dont see how that matters to the born measure.

But that a "measurement" even means in MWI, is pretty weird to me. It seems to be that what they call a measurement seems to follow from choosing an arbitrary factorisation (that does nto follow from the full state), and as the full state is assumed to evolve unitarily, it surely must evolve "as if" ALL the "measurements" defined via hte atribtrary factorisation into subsystems, apparatouse etc do happen, in the sense of all terms needs to keps in the facotirsation to get the unitary evolution. But to me this is a "werid construct", so the qustion i wonder, does any measurement happen at all? [relative to the implied perspective of MWI, ie the full state that is] as the only thinkg that "forces" this, isnt that that arbitrary factorization? And adding the factorization seems to me to "add information". And from my "inferential information perspective" this is just weird things, werid questions. It feels to me at least, conceptually inconsistent, its like one is mixing contexts at will. I doubt I will ever be a mwi fan

/Fredrik

 

[Fra]

with the "geometrisation" by assuming that independent means orthogonal are highly non-trivial assumptions. This is a

Typo. Sloppy writing again. It should say mutually exclusive not independnent.

/Fredrik

 

[Roberto Pavani]

But you're preventing that by confining everything to a narrow tube after the aperture. In wave terms, the tube would act like a waveguide.

You're partially correct. The idea was that the tube walls are coated with absorbing material, so it is not intended to behave as a conventional waveguide with reflecting boundaries.

However, your comment highlights a potential complication. Perhaps a simpler version of the thought experiment would be to remove the tube entirely and place a small detector or screen further downstream, centered on the propagation axis, so that only particles arriving near the middle are recorded.

My main question is not really about the tube itself, but about what happens after selecting the central interference maximum: would electrons behave in the same way as photons and exhibit diffraction from the aperture, or would something else be observed?

 

[PeterDonis]

Wether some "actual measuremnts" are made at all, I dont see how that matters to the born measure.

It matters because the reason for wanting a measure with the properties Gleason's Theorem requires is to be a probability measure for measurement outcomes. But that concept only makes sense if measurements have single outcomes. If all possible outcomes of a measurement happen, as in the MWI, asking what the "probability" is of a particular outcome makes no sense.